Teaching Physics with the Physics Suite Edward F. Redish

Dollar and Penny

A student makes the following argument: "I can prove a dollar equals a penny. Since a dime (10 cents) is one-tenth of a dollar, I can write:

10¢ = 0.1 \$

Square both sides of the equation. Since squares of equals are equal,

100 ¢ = 0.01 \$.

Since 100 ¢ = 1 \$ and 0.01 \$ = 1 ¢ it follows that 1\$ = 1 ¢."

What's wrong with the argument?

Solution

What's wrong is the student hasn't actually squared both sides because he forgot to square the units. A "square cent" is equal to 10-4 "square dollars", whatever that means. Think for example about the statement that 12 inches = 1 foot. If we square this relationship we get 144 square inches = 1 square foot. This is clearly the case as we can see by counting the number of square inches available in the area of 1 square foot. In the case of the penny and the dollar, the square unit has no obvious useful meaning. It just helps to warn us that the calculation does not mean what we might think it does at first.